Flight Dynamics Prediction for Scaled Mars Rotorcraft

Flight Dynamics Prediction for Scaled Mars Rotorcraft

Allen Ruan Tove ˚ Agren Aerospace Engineer Aerospace Engineer Analytical Mechanics Associates Analytical Mechanics Associates NASA Ames Research Center NASA Ames Research Center Moffett Field, CA, U.S.A. Moffett Field, CA, U.S.A.

ABSTRACT The use of sub-scale vehicles as a means of predicting full-scale vehicle behavior has historically been applied to flight dynamics testing and evaluation for aircraft operating in Earth atmospheric conditions. However, the use of sub-scale testing on Earth has not been as thoroughly explored for Martian rotorcraft. In this paper, sub-scale vehicles of varying sizes were developed in simulation using Froude scaling laws to evaluate their ability to estimate full- scale linear dynamics for the Mars hexacopter, Chopper. Blade loading, Lock number, and flap frequencies were held fixed when scaling and corresponding relationships for vehicle length, mass, inertia, and rotor speed derived.

Full-scale frequency response, gain margin, and instability characteristics are explored for hover and forward flight cases in a variety of Mars-to-Mars and Earth-to-Mars conditions. Mach effects are also analyzed as a consequence of Froude-scaling by comparing sub-scale vehicles that are Mach-matched to the full-scale Chopper. Finally, a first-order approach for downselecting a sub-scale vehicle based on feasibility is introduced.

NOTATION u , v , w Body frame translational velocities, m/s V Rotor speed at a given radial station, m/s a Speed of sound, m/s V Rotor tip speed, m/s tip A Rotor disk area, m V Airspeed, m/s ∞ C Rotor thrust coefficient, T / ( ρ A ( Ω R ) ) T X , Y , Z Stability/control derivatives for C / σ Blade loading T longitudinal, lateral, and vertical translations c Blade chord, m δ Vehicle control input, % Fr Froude number, V / ( gL ) γ Lock number, ρ acR / I tip b g Gravity, m/ s μ Advance ratio, V cos θ / ( Ω R ) ∞ G Ratio of aerodynamic to vehicle inertia, μ Dynamic viscosity, N · s/m ρ aN cR / ( 2 m ) ν Rotating natural flap frequency, rad/s b β EI Blade flapwise bending stiffness ρ Density, kg/m I Moment of inertia, kgm σ Rotor solidity, N c / ( π R ) b R R I Blade inertia, r mdr ω Frequency, rad/s or Hz b L Length, m Ω Rotor rotational speed, rad/s L , M , N Stability/control derivatives for roll, pitch, yaw φ , θ , ψ Euler angles, rad m , M Mass, kg Subscripts M Blade tip Mach number, V / a tip tip N Length scaling factor, N = L / L f s 0 . 75 75% span N Number of blades b b Blade N Number of rotors r f Full-scale p , q , r Body frame angular velocities p Lateral R Rotor radius, m q Longitudinal Re Chord-based Reynolds number, ρ V c / μ r Yaw RPM Rotor rotational speed, rev/min s Sub-scale T Time, s w Heave T Thrust, N INTRODUCTION Presented at the Vertical Flight Society’s 81st Annual Forum & Technology Display, Virginia Beach, VA, USA, May 20–22, 2025.

Following the success of NASA’s Ingenuity Mars Helicopter, This is a work of the U.S. Government and is not subject to copyright protection. next-generation Martian rotorcraft are being designed to fly farther, faster, and carry out independent mission and science for the Mars Science Helicopter in Earth and Mars environ- tasks. To provide sufficient power and maneuverability, pro- ments without scaling, noting the difficulties of designing and posed concepts, such as the Mars Science Helicopter (MSH) controlling a dynamically-matched surrogate helicopter.

and Chopper, are planned to be substantially larger than In- This paper examines the design and analysis of Froude-scaled genuity’s 1.8kg, 1.2m footprint; MSH is approximately 18kg, sub-scale vehicles as a means of predicting full-scale Mars ro- 3.9m while Chopper is approximately 33kg, 3.4m (Refs. 1,2).

torcraft linear dynamics through simulation. First, sub-scale As a result, experimental testing of the full vehicle will prove vehicles of various lengths are compared to determine the pre- to be more challenging with space limitations in current low- dictive accuracy of Froude-scaling at a constant Martian den- pressure test facilities. In particular, using methods such as sity and gravity. Second, sub-scale vehicles are simulated system identification (sys-id) (Ref. 3) to experimentally val- in Earth gravity but Mars density to predict vehicle behav- idate (linear) flight dynamics models of the full vehicle for ior in Mars gravity. Finally, models are simultaneously com- control design may prove to be challenging in hover, and po- pared across various densities, gravitational accelerations, and tentially infeasible in forward flight.

length scales, with the ultimate goal of assessing if a sub-scale The flight dynamics model validation process for Ingenuity vehicle in Earth atmospheric density and gravity can be used is documented in Ref. 4, where the bare airframe dynam- to predict full-scale Mars vehicle behavior. The frequency ics of Ingenuity were obtained through a “piece-wise” system response, as well as the full and reduced order linear time- identification campaign in which the vehicle was mounted in invariant stability and control derivatives, for the sub-scale various configurations based on the derivatives being identi- models are then scaled up to compare to those of the base- fied. Forward flight was imitated through the use of a wind line full-scale model to validate the veracity of the scaling wall and a swinging arm, in which the vehicle was affixed to methodology and to study the effect of Mach number differ- the end of an arm of a test stand, which provided results of ences. The hover case is considered first, followed by forward varying quality. Two years after Ingenuity landed on Mars, flight conditions.

free flight system identification was then performed in flights 68 and 69 on the planet itself, concluding the validation pro- MODELING cess (Ref. 5). While these techniques were sufficient to val- idate Ingenuity’s models, future missions can improve upon Vehicle Characteristics the model prediction and validation methods to be more accu- rate and generalized.

As new Mars rotorcraft grow in size, performing system iden- tification on a fixed stand may prove challenging. However, free flight testing of the full-scale vehicle is not necessarily feasible either. Given space constraints in low-pressure test- ing facilities such as the 25ft. Space Simulator at NASA Jet Propulsion Laboratory (JPL) and the Planetary Aeolian Lab- oratory (PAL) at NASA Ames, free flight testing may be re- stricted to hover conditions. Forward flight tests would likely necessitate a sufficiently large wind tunnel, wind wall, or swinging arm apparatus.

Sub-scale testing may prove to be an intermediate means Figure 1. CAD rendering of the Chopper vehicle. Credit: of predicting full-scale flight dynamics behavior as testing NASA JPL.

spaces are constrained, vehicle sizes grow, and opportunities to fly on Mars remain scarce. The use of sub-scale mod- els for dynamic testing has been well documented in fixed The Chopper vehicle features six rotors, each containing 6 wing (Refs. 6–8) and rotorcraft (Refs. 9–11), especially in the blades with an increased blade radius of 0 . 675 and a higher context of wind tunnel sub-scale testing. More recently, ef- solidity of 0 . 3 compared to Ingenuity. For the current design forts to utilize smaller unmanned aerial systems (sUAS) to iteration, the airfoils are very similar to the Ingenuity airfoils.

perform sub-scale flight tests for prediction of full-scale ve- Additionally, the rotor blades are designed to be sufficiently hicle dynamics have been pursued by Mettler (Refs. 12–14) stiff, such that the rotor dynamics do not interfere with the and Ivler (Refs. 15–17). However, while the vast majority rigid body dynamics in the frequency range of the controller.

of the existing literature focuses on Earth atmospheric con- Details of Chopper’s design methodology are covered in Ref.

ditions, examining the feasibility of scaling vehicles operat- 2 and specific aspects of the blade design and wake modeling ing in Mars conditions is less explored. Scaling from Earth are covered in Refs. 19–21. The full vehicle is approximately conditions to Mars conditions is nontrivial, as the difference 33kg and spans over 3.4m tip-to-tip. A full-scale model was in atmospheric density, speed of sound, and gravity, directly developed in FLIGHTLAB, a finite element, multi-body, ro- impacts the aerodynamic forces, moments, and damping ex- torcraft modeling and analysis tool (Ref. 22). The model perienced by the vehicle. Singh (Ref. 18) covers differences consists of a rigid fuselage and six collective-controlled ro- between the frequency response and open loop characteristics tors with rigid blades. Flapping dynamics are modeled with   hinges, tuning hinge spring stiffness and damping to emulate X X X X δ δ δ δ q p w r the first physical flapping mode. A CAD rendering of the ve-   Y Y Y Y  δ δ δ δ  q p w r hicle is shown in Fig. 1 and the corresponding FLIGHTLAB   Z Z Z Z  δ δ δ δ  q p w r model is shown in Fig. 2.   L L L L   δ δ δ δ q p w r   G = (2)  M M M M  δ δ δ δ q p w r     N N N N δ δ δ δ q p w r     0 0 0 0     0 0 0 0 0 0 0 0 Similar to most rotorcraft, Chopper is inherently unstable in open loop. For example, in hover, the longitudinal speed sta- bility derivative ( M ), a derivative characterizing sensitivity to u edgewise flow, governs the static stability of the vehicle and directly influences the frequency of unstable modes. The pitch Figure 2. FLIGHTLAB model of the Chopper vehicle.

damping derivative ( M ) also drives the magnitude of the in- q stability, dominating over other terms such as the longitudinal Rotor aerodynamic forces and moments are calculated using speed damping ( X ). However, in forward flight, coupling of u Blade Element Theory while the induced velocity is derived the lateral-longitudinal and vertical dynamics creates an in- from a three-state Peters-He wake model detailed in Ref. 23.

creased sensitivity to pitch moments due to vertical speed per- While rotor-rotor interactions are crucial to better understand- turbations, creating an angle of attack instability as a positive ing the complex aerodynamic effects of the multirotor config- M . Given that this pitch-heave instability grows at higher ad- w uration, they are not included in the modeling of this paper but vance ratios, it is of particular concern within control design are analyzed in Ref. 20.

and is explained in more detail in Ref. 24 and explicitly for Mars rotorcraft in Refs. 4, 25.

Parametric Model METHODOLOGY Creating a representative sub-scale model stipulates that the Robust control design and stability analysis is incumbent on ratio of the sub-scale vehicle’s governing forces (i.e. aerody- obtaining sufficiently accurate mathematical models of the ve- namic, gravitational, and inertial) remain consistent with the hicle. Full and reduced order linear models are generated in full-scale’s (Refs. 9,14). Nondimensional parameters, such as FLIGHTLAB at a trim point using a perturbation method, the Froude, Mach, Reynolds, and Lock numbers, relate these which calculates partial derivatives of the residuals of gen- forces such that vehicles of different sizes can be approxi- eralized equations with respect to both states and inputs and mately compared. For example, matching the Froude number are then averaged azimuthally.

ensures the ratio of inertial to gravitational forces stays con- The full order linear model for Chopper contains 108 states sistent between the sub-scale and full-scale vehicles. How- encapsulating rigid body, inflow, and flap dynamics. For con- ever, not all nondimensional terms can be adhered simultane- trol design, a quasi-static reduced order linear model can be ously (Ref. 26). For example, since both the Froude number obtained in the form M ˙ x = Fx + Gu , where x is the state vector and Reynolds number relate velocity to length scales indepen- containing the rigid body states x = [ u ; v ; w ; p ; q ; r ; φ ; θ ; ψ ] and dently, scaling with one would invariably violate the other.

u is the mixed input vector of directionally-aligned control in- Within flight dynamics, scaling the vehicle based on a consis- puts u = [ δ ; δ ; δ ; δ ] . For body-frame dynamics, the sign tent Froude number has been extensively validated in litera- q p w r convention is positive x forwards, positive y to the right, and ture and test programs, including, but not limited to, in Refs.

positive z downwards. M denotes the diagonal mass matrix, 12, 13, 15, 27, and will be the primary means of scaling within containing vehicle mass and inertias. Assuming small vehicle this paper. Additional nondimensional parameters, such as trim angles, the corresponding stability and control derivative the blade loading, are also scaled. The cumulative result is matrices are defined in Eq. (1) and Eq. (2), respectively. a methodology that simultaneously scales length components (e.g. rotor radius, hub-to-hub distance, chord), mass compo- nents (vehicle and blade mass), inertial components (vehicle   X X X X X − w X + v 0 − g 0 and blade inertias), speed components (e.g. rotor rotational u v w p q r Y Y Y Y + w Y Y − u g − g θ φ 0  u v w p q r  speed), and flapping components (e.g. hinge damping and   Z Z Z Z − v Z + u Z − g φ − g θ 0  u v w p q r    stiffness).

L L L L L L L L L u v w p q r φ θ ψ     F = M M M M M M M M M u v w p q r φ θ ψ     N N N N N N N N N u v w p q r φ ψ θ   Sub-scale Modeling  0 0 0 1 φ θ θ 0 0 0    0 0 0 0 1 − φ 0 0 0 In the context of scaled modeling, N is conventionally used to 0 0 0 0 φ 1 0 0 0 (1) represent the scale factor, defined in Eq. (3), as the ratio of the characteristic length of the full-scale to the sub-scale vehicle. vary since the rotor speeds are governed by Froude scaling, For example, if N = 2, the hub-to-hub distance of that sub- and if scaling from Earth to Mars, the speed of sound will M tip , s scale would be exactly 1/2 of the hub-to-hub distance of the differ. Taken together, the Mach number scales by = M tip , f q full-scale vehicle.

a 1 g f 1 s √ √ , decreasing at a rate of as sub-scale sizes g a N f s N 1 shrink in the same atmosphere and gravity. When compar- L = L (3) s f ing Mach from Earth to Mars for sub-scales close to the full- N scale size, the difference in gravitational acceleration domi- For a given length ratio and gravitational acceleration, Froude nates, leading to a higher Mach than on Mars. Conversely, scaling establishes a scaling relationship for speed, length, Earth sub-scales have a significantly lower Mach when the and time parameters. Rotor speeds and forward flight speeds sub-scale size shrinks and speed of sound grows.

are Froude scaled such that simulations are compared at the same advance ratio ( μ ). Vehicle mass is then scaled based on Similarly, Reynolds number can not be matched since Froude density and length-scaling (and hence, volume). Done prop- formally relates rotor speed and length scales, and for Earth- Re s erly, the blade loading ( C / σ ) is thereby maintained to ensure T to-Mars scaling, the Reynolds number scales by = Re f q each radial station is experiencing consistent lift. By exten- μ 1 g ρ f s s √ . At a constant density, gravity, and dynamic g ρ μ sion, the rotor solidity of all sub-scale vehicles is maintained N N f f s viscosity, the length difference of smaller vehicles (higher at 0.3, since both the chord and span of the blade are length- − 3 / 2 N ) scales as a function of N , decreasing Reynolds num- scaled. Lock number ( γ ) and G , a Lock number equivalent, ber dramatically when Froude scaling (an 1/8th-scale vehi- defined in Ref. 24 as a representative ratio of aerodynamic cle has a Reynold’s number of ≈ 5% the full-scale Chopper).

to vehicle inertial forces, are used to scale the blade inertia For Earth-to-Mars scaling, the Reynolds number reduction is and vehicle inertia, respectively. Finally, the flap stiffness is countered by the ratio of air densities, allowing for more simi- scaled to match the flap frequency. An overview of the scal- lar Reynolds values for smaller vehicles at higher atmospheric ing methodology can be found in Table 1, where L indicates densities on Earth and drastically higher Reynolds for larger a length unit, M indicates a mass unit, and T indicates a time vehicles.

unit. For example, to obtain a sub-scale vehicle’s mass, the full-scale vehicle’s mass is multiplied by the corresponding However, given the simulation-based nature of this work, one scale factor. Note that, while conventional tables in existing can artificially match the Mach number of the full-scale ve- literature feature most Froude-scaled parameters solely as a hicle by manipulating the speed of sound to offset the Mach function of the length-scaling factor N , Earth-to-Mars scaling reduction from length and speed scaling to assess the impact must account for differences in gravity and density and are of Mach effects. Runs that adjust the speed of sound will be derived appropriately.

termed as “Mach-matched.” While one can similarly do this to match Reynolds number by artificially prescribing the dy- namic viscosity, the airfoil tables used for these simulations Table 1. Scale Factors for Key Vehicle Properties.

were generated for a fixed density of ρ = 0 . 01 kg / m and dy- Property Dimensions Scale Factor Matching − 5 2 namic viscosity of μ = 1 . 46 × 10 N s / m . As a result, while Length L - N ρ important, effects from varying Reynolds numbers on airfoil s 1 Mass M ρ , L ρ N f q performance will not be the focus of the analysis in this cur- √ g 1 s Frequency N Fr rent paper.

T g f q L 1 g s √ Velocity Fr T g N f ρ 2 s 1 Inertia ML γ , G Predicting Full-scale Dynamics ρ N f ρ g ML s s 1 Stiffness ν 2 4 β ρ g T f f N While the sub-scale vehicle is dynamically similar to the full- scale vehicle, the sub-scale vehicle is not dimensionally sim- It is important to note that some of the simultaneous scal- ilar to the full-scale vehicle, meaning that a direct compari- ing laws presented may be practically infeasible. For ex- son of frequency responses may be misleading. For example, ample, maintaining the same C / σ in higher Earth densi- T given the nature of the length ratios, the frequency at which ties increases the Earth sub-scale vehicle mass proportionally modes occur do not inherently align. As a result, the dynam- to the increase in atmospheric density. Similarly, matching ics are “scaled up” to the full-scale vehicle to compare the flap response assuming the same material density for blades predicted dimensional stability and control derivatives, which is highly intractable for a sub-scale vehicle and would po- are relevant for full-scale Chopper control design. In the con- tentially require using different material to maintain both the text of this paper, this will be referred to as “inverse-scaling,” proper stiffness and density to satisfy both the flap frequency and requires multiplying each derivative in the linear models and Lock number matching. These feasibility concerns are by the appropriate factor. The scale-up factor for each deriva- explored at the end of the results.

tive is the inverse of the corresponding scale-down method Inevitably, Mach and Reynolds numbers will not remain con- featured in Table 1 and is shown in Table 2 for each derivative stant across scales using this methodology. Mach number will in the reduced order model. Note that the density scaling does not show up since the mass terms are separate in this repre- Notably, both the Mach number and Reynolds number vary sentation. The same dimensional analysis methodology can substantially with the relative vehicle length scale, despite in- be extrapolated to scale-up the full order linear stability and creasing higher rotor speeds at smaller sub-scale sizes. The derivatives. advancing tip Mach number disparity between the full-scale and sub-scale vehicles also grows as the advance ratio in- creases. While not included in the table, other parameters Table 2. Scale Up Factors for Dimensional Derivatives.

such as the vehicle inertias, blade mass distribution, flap hinge Derivative Dimensions Scale Up Factor stiffness were also scaled accordingly, based on the method- X , X , X u v w ology outlined in Table 1.

Y , Y , Y u v w q All vehicles were trimmed and linearized at hover and ad- g f 1 1 √ Z , Z , Z u v w T g vance ratios of 0.056, 0.111, and 0.165 (equivalent to full- N s L , L , L scale forward flight speeds of 10, 20, and 30 m/s, respec- p q r M , M , M tively), with and without Mach number matching. Fig. 3 p q r N , N , N shows the bare-airframe pitch response from 0.01 to 150 Hz p q r X , X , X for three sub-scale vehicles in hover without inverse scaling p q r q √ g L f to full Chopper dimensions or Mach-matching. Decreasing Y , Y , Y N p q r T g s the length-scale, and hence increasing the rotor speed from Z , Z , Z p q r Froude-scaling, pushes the phugoid and blade flapping modes X , X , X , X δ δ δ δ q p w r to higher frequencies but the same per/rev (for example, the g L f Y , Y , Y , Y δ δ δ δ 2 q p w r g flap mode occurs at roughly 1.3/rev for all vehicles).

T s Z , Z , Z , Z δ δ δ δ q p w r L , L , L u v w q g f 1 1 √ M , M , M u v w LT g s N N N , N , N u v w L , L , L φ θ ψ M , M , M ψ φ θ g f 1 1 N , N , N φ θ ψ N g T s L , L , L , L δ δ δ δ q p w r M , M , M , M δ δ δ δ q p w r N , N , N , N δ δ δ δ q p w r RESULTS To validate the feasibility of the scaling methodology, sub- scale vehicles are evaluated in their ability to predict full-scale Chopper dynamics under progressively less constraining as- sumptions. First, length scales are varied at the same Martian Figure 3. Full order q / δ frequency response for μ = 0 q environmental conditions (atmospheric density and gravity).

without inverse-scaling or Mach-matching.

Next, sub-scale vehicles are compared at the same density but different gravity and length scales. Finally, sub-scale vehicles Fig. 4 shows the same frequency response inverse-scaled to of simultaneously varying lengths, gravity, and atmospheric the full-scale vehicle for a pitch on-axis response in hover, density are compared, cumulating in a test matrix of 72 dif- yet without Mach-matching. By applying the inverse scaling ferent sub-scale vehicle scenarios. A preliminary approach laws, the sub-scale vehicles better predict the phugoid and flap is then taken to downselect potential candidates for sub-scale frequencies of the full-scale vehicle accurately. From here on, testing based on various feasibility criteria.

all figures will be shown to be inverse-scaled up for direct comparison to the full-scale vehicle dynamics. However, the smaller sub-scale vehicles, notably the 25% sub-scale vehi- Varying Length, Fixed Density and Gravity cle, seem slightly less damped in the magnitude around the Seven sub-scale vehicles of different length scales were mod- phugoid mode. This damping is primarily a result of the dif- eled in reference to the Chopper full-scale vehicle in FLIGHT- ference in Mach effects, as the rotors for the sub-scale vehicles 3 2 LAB for ρ = 0 . 012 kg / m and g = 3 . 71 m / s . Table 3 out- operate at a lower Mach number compared to the full vehi- lines key nondimensional terms and modeling parameters, cle (0.39 Mach for the 25% vs. 0.77 Mach for the full-scale where the percentage indicates the sub-scale length relative vehicle). The mitigation of Mach effects is shown in Fig. 5, to the full-scale vehicle. Across all vehicles, the Froude and where the tip Mach numbers were matched by artificially scal- Lock number, solidity, and blade loading are kept constant. ing the speed of sound, and the attenuation disappears for the 3 2 Table 3. Parameters of Sub-scale Vehicles vs. N at ρ = 0 . 012 kg / m and g = 3 . 71 m / s .

100% 87.5% 75% 62.5% 50% 37.5% 25% 12.5% N 1 8/7 4/3 8/5 2 8/3 4 8 R [m] 0.675 0.59 0.51 0.42 0.34 0.25 0.17 0.08 Tip-to-Tip Distance [m] 3.43 3.00 2.57 2.14 1.71 1.29 0.86 0.43 Total Mass [kg] 33.0 22.1 13.9 8.1 4.13 1.74 0.52 0.06 RPM 2540 2715 2932 3212 3591 4147 5079 7183 γ 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 Hover, μ = 0 x Fr 12868 12868 12868 12868 12868 12868 12868 12868 C T 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 σ M (Not Ma-matched) 0.77 0.72 0.67 0.61 0.55 0.47 0.39 0.27 tip M (Ma-matched) 0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.77 tip Re 12300 10100 8000 6100 4400 2800 1500 500 0 . 75 μ = 0 . 111 x V [m/s] 20.0 18.7 17.3 15.8 14.1 12.2 10.0 7.07 ∞ Fr 15895 15895 15895 15895 15895 15895 15895 15895 C T 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 σ M (Not Ma-matched) 0.86 0.80 0.74 0.68 0.60 0.52 0.43 0.30 tip M (Ma-matched) 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 tip Re 14200 11600 9200 7000 5000 3300 1800 630 0 . 75 predicted frequency responses. The full order frequency re- sponses show fantastic alignment after being Mach-matched, validating the scaling process.

Figure 5. Full order q / δ frequency response for μ = 0 q with inverse-scaling and Mach-matching.

in the full-scale Chopper Mach number in hover (M > 0 . 77) tip Figure 4. Full order q / δ frequency response for μ = 0 q while the dashed lines correspond to hover tip Mach num- with inverse-scaling but without Mach-matching.

bers lower than 0 . 77. From the figure, it can be seen that the lower the sub-scale tip Mach number, the more attenuated the The attenuation (and amplification) from Mach effects can gain of the longitudinal phugoid mode will be, matching the be seen in Fig. 6, which enlarges the longitudinal phugoid previous results seen in Fig. 4. More specifically, the 80% mode for a 50% sub-scale vehicle in Mars density, showing Mach-matched line corresponds to an overprediction in gain the impact as a function of the severity of Mach number de- margin of about 9%, or around 0 . 28dB. Conversely, larger ex- viation. The red line indicates the full-scale Chopper linear ceedances in sub-scale tip Mach amplify the magnitude of model (M = 0 . 77) and grayscale lines are for the 50% sub- the mode, where a 20% overshoot causes an overprediction tip scale’s predicted full-scale linear model at varying degrees of in gain margin by around 1 . 4%, or 0 . 09dB. Note that this Mach-matching. The dotted lines correspond to exceedances may be valid in subsonic conditions and that substantive ex- ceedances in the tip Mach number increasingly creep into the high sub-sonic regime (especially for forward flight) may in- validate such behavior.

Figure 7. Reduced order longitudinal open loop gain mar- gin vs. forward flight speeds and Mach-matching.

Figure 6. Magnitude variation of phugoid mode for q / δ q at μ = 0 from Mach effects. % indicates how close the sub- scale tip Mach number is relative to the full-scale Chopper tip Mach for the same condition.

The predicted reduced order open loop gain margins corrobo- rate the impact of Mach-matching, as seen in Fig. 7. In hover, the difference in the average predicted gain margin from the Mach-matched runs provided more accurate results compared to the non-Mach-matched runs (5% off the full-scale pitch re- sponse gain margin compared to 20%). For all advance ratios except μ = 0 . 056, Mach-matching results in less dispersion Figure 8. Growing forward flight instability due to in- in the predicted gain margins. Furthermore, the sub-scale pre- creasing M for three predicted sub-scales without Mach- w dicted margins underestimate the margin at higher speeds and matching.

overestimate in hover.

As aforementioned, the pitch-heave instability that arises in The Mach-matched plot of predicted M derivatives is shown w forward flight is of primary concern. To examine the forward in Fig. 10 as a function of the N length scale. The x-axis de- flight characteristics of the sub-scale predicted linear models, lineates the size of the sub-scale vehicle used to predict the Fig. 8 presents a root-locus of four vehicles at four different derivative, and the line styles represent the various forward advance ratios without Mach-matching, focusing primarily on flight speeds. The red curves indicate the Mach-matched val- the evolution of the longitudinal and lateral phugoid modes. ues and the light blue curves indicate the non-Mach-matched The colors indicate the sub-scale vehicle used to predict the values. To determine if M can successfully be predicted from w full-scale behavior, and the poles of all four advance ratios are any sub-scale length, a dotted horizontal line at the value of co-plotted. Both the Mach and non-Mach-matched cases re- the full-scale (100%) Chopper’s M derivative is drawn. As w veal the same trend of an increasingly unstable mode caused a result, close adherence to the dotted line across sub-scale by the growing angle of attack derivative. Furthermore, the sizes shows up as a horizontal line at the value of the dot- smaller the sub-scale vehicle used to predict the full-scale, ted 100% Chopper line. While the non-Mach-matched M w the more damped each pole is. When Mach effects are cor- predicts relatively well for lower advance ratios, there is a rected, as seen in Fig. 9, the predicted full-scale poles align growing deviation as the forward flight speed increases and very closely with the full-scale Chopper poles. The remaining the length scale decreases (25% difference for a 12 . 5% vehi- discrepancies are likely attributed to numerical differences in cle at μ = 0 . 165). Since the difference in Mach numbers for the simulation process. the smallest length scales is nontrivial (0 . 3 versus 0 . 9 for the the full-scale Chopper grows, leading to increasing underpre- dictions of the gain margin in the longitudinal axis, and under- predictions of the angle of attack instability derivative. These differences are primarily due to Mach effects, and can be cor- ) -1 rected once the tip Mach number is matched. As a result, by testing a sub-scale vehicle at the same gravity and atmospheric density (e.g. building a sub-scale to test in a sufficiently large vacuum chamber with a responsive gravity offload system), one could theoretically predict the full-scale dynamic behav- ior, capturing key dynamics in hover and forward flight.

Varying Length and Gravity, Fixed Density Imaginary Axis (seconds However, vacuum testing is often expensive, time- constraining, and limited in availability. Therefore, it is of interest to examine if the dynamics of Mars rotorcraft can -1 be predicted from simplified testing environments without the Real Axis (seconds ) need for a gravity offload system. To that end, sub-scale mod- Figure 9. Pole-zero prediction from a 50% sub-scale for els in Mars density but Earth gravity ( g = 9 . 81 m / s ) were μ = 0 . 165 with Mach-matching.

simulated, and key parameters for four sub-scale vehicles are highlighted in Table 4. Since Froude number is a function of same conditions), these deviations are remedied when Mach gravity, rotor speeds and forward flight speeds increase by a q effects are corrected, as seen by the approximately flat red line g e factor of ≈ 1 . 6 for sub-scale vehicles when simulating g for the Mach-matched case. m from Earth gravity. Critically, despite Earth’s higher speed of sound at 1atm, the tip Mach number also increases q g a e m by approximately ≈ 11% for each trim scenario.

g a m e Changing RPM also affects flap frequency, requiring scaling of the flap hinge stiffness if matching the flap response of the vehicle is desired. However, length scales, mass, Froude number, Lock number, and blade loading all remain the same as the prior Mars gravity case.

Mars =0.012, 100% Earth =0.012, 100% Earth =0.012, 75% Earth =0.012, 50% Earth =0.012, 25% Figure 10. M from sub-scales vehicles of different lengths w at ρ = 0 . 012 kg / m .

In conclusion, the full-scale linear dynamics can be predicted from sub-scale vehicles of varying lengths in Mars gravity and atmospheric density, especially for hover and lower ad- vance ratios. Smaller sub-scale vehicles do bring poles closer to the origin and attenuate the full order longitudinal fre- quency response compared to the full-scale vehicle. While re- sults for only the longitudinal axes is shown, other axes show Figure 11. Full order q / δ frequency response for μ = 0 q similar results. Similarly, while M was predicted well, the with inverse-scaling but without Mach-matching for sub- w pitch damping derivative M , was predicted with less accu- scales in Earth and Mars gravity.

u racy. However, this did not significantly impact the predicted full-scale frequency response and gain margin, as seen by the The full order pitch response in hover between inverse- alignment. Furthermore, as advance ratios increase, the differ- scaled sub-scale vehicles in Earth gravity and Mars density ence in Mach number between the smallest scale vehicles and ( ρ = 0 . 012 kg / m ) without Mach-matching and the full-scale Table 4. Parameters of Sub-scale Vehicles vs. N at ρ = 0 . 012 kg / m and Earth gravity.

Mars Earth Earth Earth Earth 100% 100% 75% 50% 25% Total Mass [kg] 33.0 33.0 13.9 4.13 0.52 RPM 2540 4129 4768 5839 8258 γ 0.98 0.98 0.98 0.98 0.98 Hover, μ = 0 x Fr 12868 12868 12868 12868 12868 C T 0.126 0.126 0.126 0.126 0.126 σ M (Not Ma-matched) 0.77 0.94 0.82 0.67 0.47 tip Re 12300 18800 12200 6600 2300 0 . 75 μ = 0 . 111 x V [m/s] 20.0 32.5 28.2 23.0 16.3 ∞ Fr 15895 15895 15895 15895 15895 C T 0.1 0.1 0.1 0.1 0.1 σ M (Not Ma-matched) 0.86 1.05 0.91 0.74 0.52 tip Re 14200 21600 14000 7600 2700 0 . 75 Chopper is shown in Fig. 11. Note that the flap frequency has higher forward flight speeds, the frequency of vehicle and ro- been matched by scaling the blade bending stiffness, modeled tor modes are captured from the various sub-scales, but the by scaling the hinge stiffness. Aside from variations in the magnitude is not fully captured even when Mach effects are magnitude of the frequency regime below 1 rad/s, the general corrected, as seen in Fig. 13. The lower frequency deviations response across all sub-scale lengths is very similar. As in are potentially attributable to numerical discrepancies in the the Mars gravity case, the amplitude of the lower frequency inverse-scaling of the inflow states which are removed in the response is susceptible to Mach effects, where sub-scale vehi- model reduction. This can be seen in Fig. 14, as the reduced cles with tip Mach numbers greater than the full-scale vehicle order pitch response in forward flight shows significantly bet- overshoot the gain and ones with lower tip Mach numbers are ter adherence, even without Mach-matching.

more attenuated, with the exception of the 100% Earth grav- ity vehicle. This sub-scale vehicle has a high sub-sonic tip Mach of 0.94 in hover, which resulted in a severely atten- uated phugoid response. These variations in magnitude are corrected in the Mach-matched case, shown in Fig. 12.

Mars =0.012, 100% Earth =0.012, 100% Earth =0.012, 75% Earth =0.012, 50% Earth =0.012, 25% Mars =0.012, 100% Earth =0.012, 100% Earth =0.012, 75% Earth =0.012, 50% Earth =0.012, 25% Figure 13. Full order q / δ frequency response for μ = 0 . 11 q with inverse-scaling and Mach-matching for sub-scales in Earth and Mars gravity.

The reduced order gain margin for different advance ratios Figure 12. Full order q / δ frequency response for μ = 0 q with and without Mach-matching is shown in Fig. 15. In with inverse-scaling and Mach-matching for sub-scales in hover, the Earth gravity sub-scale models without Mach- Earth and Mars gravity.

matching generally overpredict by 0 . 8dB on average, where Complications arise when comparing in forward flight. At decreasing sub-scale size correlates to larger overpredictions.

the Earth gravity case, the methodology remains consistent, Mars =0.012, 100% especially for reduced order model estimation.

Earth =0.012, 100% Earth =0.012, 75% Earth =0.012, 50% Earth =0.012, 25% Figure 14. Reduced order q / δ frequency response for q μ = 0 . 11 with inverse-scaling but without Mach-matching Figure 16. M from sub-scale vehicles of different lengths w between Earth and Mars gravity sub-scales.

and gravitational accelerations at ρ = 0 . 012 kg / m .

The trend reverses for advance ratios of 0 . 11 and 0 . 165, where the smaller sub-scales tend to underpredict the gain margin the most, and larger sub-scales overpredict. There is improve- Varying Length, Gravity, and Density ment in gain margin prediction when Mach-matching, espe- Full Earth to Mars scaling requires varying density, in addi- cially for the hover and μ = 0 . 165 cases.

tion to gravity, and length-scales. While density variations do not directly impact Froude number, they do impact the thrust coefficient, C , which consequently affects mass, inertia, and T flap hinge stiffness/damping scaling. Nine densities, rang- ing from ρ = 0 . 012 to 1 . 225 kg / m , in addition to the eight length-scales, were selected to create sub-scale vehicles that were trimmed and linearized in Earth gravity ( g = 9 . 81 m / s ).

All were simulated with and without Mach-matching, as well as at the 4 advance ratios, resulting in over 500 unique scaled linear models. To simplify comparisons, the “mass ratio,” de- ρ m 1 s s fined as = , is used as a comparative heuristic since it m ρ N f f is a function of both the atmospheric density and length scale.

Intuitively, it relates the sub-scale mass to the mass of the full-scale Chopper. Table 5 outlines the key non-dimensional parameters for a few sub-scale vehicles of the same size at varying densities. The blade loading, Lock number, Froude number, and advance ratios remain matched. Notably, the Reynolds number grows dramatically as the density increases.

The full-scale Chopper’s longitudinal frequency response in Figure 15. Reduced order longitudinal open loop gain hover remained very well predicted across density variations margin vs. advance ratio and Mach-matching.

even without Mach-matching. Fig. 17 highlights the differ- To conclude the low atmospheric density, Earth gravity anal- ence in predicted full-scale longitudinal gain margin for the ysis, the Mach-matched inverse-scaled angle of attack deriva- reduced order linear model in hover. For simulations that were tive, M , is shown in Fig. 16. While in ideal environmen- not Mach-matched, larger sub-scale vehicles ( ≥ 87 . 5%) pre- w tal conditions, this figure predicts that even with a different dicted the margins within 2% of the full-scale at all densi- gravity, this derivative can be estimated given any size sub- ties other than ρ = 1 . 225 kg / m . As the sub-scale size de- scale vehicle. The real-world analog would entail Earth test- creases, the predicted gain margin is increasingly overesti- ing in either a vacuum chamber without a gravity offload sys- mated, as seen in both previous Mars and Earth gravity cases.

tem or outside at a sufficiently high altitude. The results show Conversely, the Mach-matched cases provided consistent gain that while the discrepancies in gain margin and full order fre- margin overpredictions ≈ 3 − 5% for the vast majority of sub- quency response are larger at higher forward flight speeds for scale size and density combinations. The highest atmospheric Table 5. Parameters of Sub-scale Vehicles vs. N at Varying Densities and Earth gravity.

Mars Earth Earth Earth Earth Earth 100% 75% 75% 75% 75% 75% ρ [kg / m ] 0.012 0.012 0.358 0.531 0.878 1.225 Mass Ratio 1.0 0.42 12.6 18.7 30.9 43.1 RPM 2540 4768 4768 4768 4768 4768 γ 0.98 0.98 0.98 0.98 0.98 0.98 Hover, μ = 0 x Fr 12868 12868 12868 12868 12868 12868 C T 0.126 0.126 0.126 0.126 0.126 0.126 σ M (Not Ma-matched) 0.77 0.82 0.86 0.82 0.77 0.74 tip Re 12300 12200 394600 545100 818500 1070000 0 . 75 μ = 0 . 111 x V [m/s] 20.0 28.2 28.2 28.2 28.2 28.2 ∞ Fr 15895 15895 15895 15895 15895 15895 ¯ C T 0.1 0.1 0.1 0.1 0.1 0.1 σ M (Not Ma-matched) 0.86 0.91 0.95 0.91 0.86 0.83 tip Re 14200 14000 454400 627600 942400 1232000 0 . 75 Mars =0.012, 100% Earth =0.358, 75% Earth =0.531, 75% Earth =0.878, 75% Earth =1.225, 75% Figure 17. Reduced order longitudinal open loop gain Figure 18. Full order q / δ frequency response for μ = 0 . 11 q margin in hover for various atmospheric densities, with with inverse-scaling and Mach-matching between Earth and without Mach-matching. and Mars gravity sub-scales in various air densities.

density case shows the largest dispersion in predicted gain the same behavior as the 75% sub-scale in the Earth ρ = margin, where the largest sub-scale vehicles have errors ex- 0 . 012 kg / m case in Fig. 13. Without Mach-matching, the ceeding 20%, of which the causes are not entirely resolved at frequency response (not pictured) reveals worse adherence in this time. These deviations are similarly seen in the Mach- the lower frequencies and greater variability in the higher fre- matched case. Similar trends can be found for the gain mar- quency ( > 100 rad/s) regime. These are predominantly from gin tables for the forward flight cases, except the deviation higher order terms such as the flapping and inflow, as seen in in gain margin at the smallest sub-scales is lower (around 9% the better adherence in the non-Mach-matched reduced order for μ = 0 . 165) and the differences at the largest sub-scales are response, shown in Fig. 19.

slightly higher ( ≈ 7 − 8%).

Similarly, predicting the poles at different forward flight A comparison of the predicted full-scale Chopper longitudinal speeds continues to match across various Earth atmospheric response from 75%-length sub-scales at μ = 0 . 11 is shown densities as seen in Fig. 20. As seen previously, the longitudi- in Fig. 18. The general location of the various higher or- nal phugoid mode continues to track the full-scale evolution der modes align, as expected from scaling the rotor blade with respect to higher forward flight speeds. However, there stiffness. The phugoid mode is especially more damped for is also a general trend that the Earth densities slightly under- the Earth conditions compared to the Mars case and matches predict the magnitude of the instability of the poles, but this Mars =0.012, 100% Earth =0.358, 75% Earth =0.531, 75% Earth =0.878, 75% Earth =1.225, 75% ) -1 Imaginary Axis (seconds -1 Real Axis (seconds ) Figure 19. Reduced order q / δ frequency response for Figure 21. Pole-zero prediction from a 75% sub-scale for q μ = 0 . 11 with inverse-scaling but without Mach-matching μ = 0 . 165 in various air densities with Mach-matching.

between Earth and Mars gravity sub-scales in various air Earth ρ = 0 . 012 kg / m case, there is an increasing underpre- densities.

diction of the derivative as the advance ratio increases, indi- is remedied once Mach effects are matched as seen in Fig. 21.

cated by a growing deviation from the full-scale Chopper’s Furthermore, for larger sub-scales at higher atmospheric den- values. This deviation widens as the mass ratio decreases, sities, there is a consistent exaggeration of the instability of indicating similar trends to the previous cases where smaller the poles, overpredicting the magnitude slightly at the highest sub-scales underpredicted the derivative value.

densities. Conversely, the smaller sub-scale vehicles provide a more consistent prediction with the poles.

Figure 22. M from sub-scale vehicles of different lengths, w gravitational accelerations, and atmospheric densities Figure 20. Growing forward flight instability due to in- without Mach-matching.

creasing M for a 75% sub-scale vehicle in various air den- w sities without Mach-matching.

Most of these effects are remedied when looking at the Mach- matched equivalents, where M predictions are consistent w As a result, it remains critical to see if the angle of attack across the vast majority of mass-ratios. However, the largest stability derivative ( M ) can be estimated from Earth con- w vehicles in the Earth ρ = 1 . 225 kg / m (75%, 88%, and 100%- ditions. Fig. 22 and Fig. 23 show the predicted M deriva- w size sub-scales) continue to depict a substantive deviation in tive from all environments and length-scales for all forward the full-scale prediction, similar to the gain margin analy- flight speeds without and with Mach-matching, respectively.

sis. This is found in both the non-Mach-matched and Mach- Derivatives are plotted against a logarithmic vehicle mass ra- matched cases and is an area for further investigation.

tio, where mass ratios ≫ 1 indicate a substantively larger sub- scale mass relative to the full-scale Chopper. Similar to the and atmospheric density and the columns indicate the length of the sub-scale vehicle relative to the full-scale. The bot- tom right corner represents the full-scale Mars Chopper vehi- cle in condition. Darker regions of the heatmap indicate sub- scales with a similar mass to the full-scale vehicle. Outside of the highest density case, the larger sub-scale vehicles tended to best predict the longitudinal gain margin, especially since the tip Mach numbers remained closest to the full-scale vehi- cle’s. However, many of these vehicles are high in mass, and practical considerations, such as ease of transport, favor vehi- cles that are smaller, especially at higher density atmospheric conditions. This may come at the expense of the integrity of the full-scale flight dynamics prediction, especially when Mach number is not scaled. Conversely, the smallest vehi- cles at lower densities have a significantly constrained mass budget that may struggle to accommodate heavier subcom- ponents. As a result, testing in ρ = 1 . 225 kg / m and Earth Figure 23. M from sub-scale vehicles of different lengths, w gravity likely requires smaller (e.g. 12 . 5% or 25%) sub-scale gravitational accelerations, and atmospheric densities vehicles while testing in ρ = 0 . 012 kg / m offers more flexi- with Mach-matching.

bility.

In conclusion, the scaling methodology continues to predict key longitudinal dynamics characteristics of the full-scale ve- 6.59 52.72 177.93 421.77 823.77 1423.48 2260.43 3374.16 Earth, 1.225 hicle even in varying atmospheric density. Mach-matching does provide slight improvements, especially in predicting 5.66 45.26 152.76 362.11 707.24 1222.11 1940.67 2896.86 Earth, 1.051 full-scale gain margins from smaller length scale vehicles.

4.73 37.81 127.59 302.45 590.71 1020.75 1620.92 2419.56 Similarly, vehicles that were operating close to the desired Earth, 0.878 Mach number (e.g. the 75% sub-scale at ρ = 0 . 878 kg / m ) 3.79 30.35 102.42 242.78 474.18 819.39 1301.16 1942.26 Earth, 0.705 also predicted values that were closer to the full-scale Chop- ] per’s. However, predictions from larger sub-scale sizes at the 2.86 22.89 77.25 183.12 357.66 618.03 981.41 1464.96 Earth, 0.531 highest density ( ρ = 1 . 225 kg / m ) show significant decreases [kg/m in predictive accuracy for both the longitudinal gain margin 1.93 15.43 52.08 123.46 241.13 416.67 661.65 987.66 Earth, 0.358 and the M derivative.

w 1.0 7.97 26.91 63.79 124.6 215.31 341.9 510.35 Earth, 0.185 Limitations and Feasibility 0.06 0.52 1.74 4.13 8.07 13.94 22.14 33.05 Earth, 0.012 While promising, the results shown rely on assumptions that 0.06 0.52 1.74 4.13 8.07 13.94 22.14 33.05 Mars, 0.012 may be infeasible in practice. For example, the mass estima- tion does not take into account the more granular scaling of 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100.0 individual subcomponents of the vehicle, such as the batteries Sub-scale Size [%] and motors. Potential Reynolds effects were not examined, Figure 24. Matrix of feasible vehicle mass [kg].

and the airfoil performance look-up tables were assumed to be the same at every atmospheric condition. This potentially overestimates the lift generation for the smaller sub-scale ve- Furthermore, while it is possible to match the flap frequency hicles at lower densities and underestimates their efficiency at by scaling the hinge stiffness in simulation, matching the flap significantly higher densities. Stall and near-stall predictions stiffness in practice is nontrivial, especially when simulta- are also assumed to be the same. Furthermore, part of the test neously scaling the blade inertia, and hence mass distribu- matrix of length-scales and densities contains sub-scale vehi- tion. This appears in the volumetric density of the blade ( ρ ), b I cles that are, at the minimum, physically complex to manu- b g f f which is scaled by N to match the blade flapping stiff- g I s facture, and more likely than not, practically infeasible. bs ness ( EI ). Fig. 25 provides a heatmap of the test configura- b ρ s 1 Vehicle mass scales with , indicating that smaller length- ρ tions shown and the corresponding blade volumetric density f N scales decrease vehicle mass by N , while greater atmospheric needed. The tiles that have no values require blade densities densities linearly increase the total vehicle mass. As a re- that exceed the volumetric density of steel. If the intention is sult, sub-scale vehicles ranging from 0 . 06 (lowest density and to maintain the blade density around ρ = 0 . 0015 g / mm , the b smallest size) to > 3000 kg (highest density and largest size) material density for carbonfiber, only a few vehicles would were included in the analysis. Fig. 24 shows a heatmap of provide likely options, including a 62 . 5% vehicle flown in the 72 sub-scale vehicles modeled, where each square is la- ρ = 0 . 531 kg / m . Otherwise, scaling the corresponding stiff- beled with its respective mass. The rows denote the gravity ness for an Earth atmospheric density would potentially re- quire blades to be made of higher densities such as aluminum the test environment’s speed of sound, potentially by alter- and steel. Forgoing scaling of the blade flap response is a ing the gas composition of a testing chamber such as using reasonable approach as long as the flap frequency remains refrigerant (Ref. 6) or flying at a sufficiently high altitude.

sufficiently outside the bandwidth of control and simultane- Cumulatively, designing a sub-scale vehicle that satisfies all ously ensure that the Mars blades can structurally withstand three of the above conditions, in addition to the scaling laws, the larger drag forces at greater air densities.

may be challenging. For example, a 62 . 5% sub-scale flying on Earth at a low atmospheric density of ρ = 0 . 531 kg / m has 5768.0 619.3 100.0 21.4 5.6 Earth, 1.225 a very similar tip Mach number and requires a similar blade density to Chopper, but also requires a vehicle mass 10 times Earth, 1.051 6722.9 721.9 116.6 25.0 6.6 the Chopper vehicle. As a result, it is likely that at least one of the physical parameters has to be sacrificed in order to arrive Earth, 0.878 8047.6 864.1 139.6 29.9 7.9 at a practical solution.

Earth, 0.705 1076.2 173.8 37.2 9.8 ] CONCLUSIONS Earth, 0.531 1428.8 230.8 49.4 13.0 [kg/m Sub-scale vehicles of varying sizes were developed using Earth, 0.358 2119.2 342.3 73.3 19.3 Froude scaling laws to evaluate their ability to predict full- scale dynamics of the next-generation Mars rotorcraft, Chop- Earth, 0.185 4101.0 662.3 141.8 37.3 per. Blade loading, Lock number, and flap frequencies were also matched and scaling relationships derived. The method- Earth, 0.012 2185.8 575.0 ology was tested to predict full-scale frequency response, Mars, 0.012 5777.8 1520.0 gain margin, and stability characteristics in hover and forward flight for a matrix of Mars and Earth gravitational accelera- 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100.0 tions and atmospheric densities. Generally, smaller length- Sub-scale Size [%] scale vehicles demonstrated slightly worse predictive capabil- ity primarily due to Mach effects, which could be corrected if Figure 25. Matrix of feasible blade densities [kg/m ].

the speed of sound was adjusted to match the advancing tip Mach number of the full-scale vehicle. Longitudinal effects such as the pitch-heave instability at higher forward flight 0.3371 0.4767 0.5838 0.6741 0.7537 0.8256 0.8918 0.9533 Earth, 1.225 speeds were also successfully replicated from all conditions, 0.3432 0.4853 0.5944 0.6864 0.7674 0.8406 0.908 0.9707 but were more correctly estimated if Mach effects were ac- Earth, 1.051 counted for. Sub-scale vehicles at the highest density also 0.3505 0.4957 0.6071 0.701 0.7837 0.8585 0.9273 0.9913 Earth, 0.878 suffered in predictive capability, leading to larger deviations in predicted values of the angle of attack stability derivative M w 0.3596 0.5086 0.6229 0.7193 0.8042 0.8809 0.9515 1.0172 Earth, 0.705 ] and gain margins. Limitations in vehicle mass, blade density and material, and Mach-matching capabilities constrain feasi- 0.3718 0.5258 0.644 0.7436 0.8313 0.9107 0.9837 1.0516 Earth, 0.531 bility for the full sub-scale matrix, and must be considered for [kg/m implementation, potentially necessitating the sacrifice of one 0.3887 0.5497 0.6732 0.7774 0.8692 0.9521 1.0284 1.0994 Earth, 0.358 aspect of matching. However, if accounted for, it has been 0.3887 0.5497 0.6732 0.7774 0.8692 0.9521 1.0284 1.0994 Earth, 0.185 shown that given the right conditions, a sub-scale model of varying length in a different atmospheric density and gravity 0.3705 0.5239 0.6416 0.7409 0.8284 0.9074 0.9801 1.0478 Earth, 0.012 has the potential to predict the full-scale Chopper dynamics, at least in a reduced-order state space representation.

0.3026 0.4279 0.5241 0.6051 0.6766 0.7412 0.8005 0.8558 Mars, 0.012 12.5 25.0 37.5 50.0 62.5 75.0 87.5 FUTURE WORK 100.0 Sub-scale Size [%] This paper has confirmed that Froude scaling can be applied Figure 26. Matrix of feasible Mach numbers for μ = 0 . 11 .

to create sub-scale vehicles that predict full-scale Mars ro- torcraft linear flight dynamics in simulation. However, im- Finally, results have shown that neglecting Mach effects may provements can be made to the analysis. Using dedicated air- contribute to deviations in frequency response analysis, espe- foils to properly account for Reynolds number effects for each cially for smaller sub-scale vehicles when the difference in sub-scale size and flight condition would provide more real- Mach numbers grows. Fig. 26 reveals a region of feasibility istic rotor performance estimation, which might affect trim (demarcated in dark blue) in the test matrix that has the clos- conditions and further inform which vehicles are more practi- est tip Mach number to the full-scale vehicle without the need cally feasible. Furthermore, the inclusion of higher fidelity in- for any Mach-matching at the highest advance ratio. Other- flow modeling through FLIGHTLAB’s VVPM and the inclu- wise, matching Mach numbers would require manipulating sion of rotor-rotor interference would improve the accuracy of the modeling (for example, the longitudinal speed damp- 5. Aagren, T. S., Ruan, A. W., Malpica, C., Withrow- ing derivative, X , is sensitive to longitudinal variations of the Maser, S., and Meyn, L., “In-flight System Identification u inflow). Finally, nonlinear response through system identifi- of the Ingenuity Mars Helicopter,” AIAA SciTech 2025 cation should be performed both in simulation and in exper- Forum, Orlando, FL, 2025.

imentation to validate the findings. Low pressure testing fa- 6. Wolowicz, C. H., Bowman, J. S., and Gilbert, W. P., cilities at NASA Ames and JPL may provide opportunities to “Similitude Requirements and Scaling Relationships as perform sub-scale free flight system identification for compar- Applied to Model Testing,” NASA Technical Paper ison.

1435, 1979.

Author contact: 7. Owens, B., Brandon, J., Croom, M., Fremaux, C. M., • Allen Ruan allen.w.ruan@nasa.gov Heim, E., and Vicroy, D., “Overview of Dynamic Test Techniques for Flight Dynamics Research at NASA • Tove ˚ Agren tove.s.aagren@nasa.gov LaRC,” AIAA 25th Aerodynamic Measurement Tech- nology and Ground Testing Conference, San Francisco, ACKNOWLEDGMENTS CA, June 2006.

This work was carried out under funding from the New Busi- 8. Chambers, J. R., Modeling Flight: The Role of Dy- ness Council at NASA Ames Research Center, which the au- namically Scaled Free-Flight Models in Support of thors thank for their resources and support. The authors would NASA’s Aerospace Programs , Government Printing Of- like to express their gratitude to Carlos Malpica, Wayne John- fice, 2009. DOI: NASA SP 2009-575.

son, David Caudle, Jeremy Aires, Witold Koning, Nicholas Peters, Carlos Pereyra, Gianmarco Sahragard-Monafred, and 9. Curtiss, H., Putman, W., and Martinez, E., “The Eval- Sesi Kottapalli for their technical expertise and editorial feed- uation of Stability and Control Characteristics of Air- back. Furthermore, the authors thank Lucas Rowden and Jett craft at Low Speeds Using Dynamically Similar Models Wong-Parker, Rotorcraft Aeromechanics Summer 2024 in- in Semi-Free Flight,” American Helicopter Society 18th terns, for their early contributions to the project. Finally, the Annual Forum, Washington, DC, May 1962.

authors would like to acknowledge the support of Shannah Withrow-Maser, Carl Russell, Dr. William Warmbrodt, and 10. Hunt, G. K., “Similarity Requirements for Aeroelastic Larry Hogle during this research. Models or Helicopter Rotors,” Royal Aircraft Establish- ment C.P. 1245, 1972.

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